In probability theory and statistics, a normal variance-mean mixture of the form

are independent is normally distributed is continuously distributed on the positive half-axis with probability density function. The conditional distribution. A normal variance-mean mixture can be thought of as the distribution of a certain quantity in an inhomogeneous population consisting of many different normal distributed subpopulations. It is the distribution of the position of a Wiener process. An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution.

The probability density function of a normal variance-mean mixture with mixing probability density is

and its moment generating function is

, i.e.

See also

• Normal-inverse Gaussian distribution

• Variance-gamma distribution

• Generalised hyperbolic distribution

References

• O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", International Statistical Review, 50, 145–159.